Finally, it is tempting to speculate on the ramifications of these codes for
self-correcting quantum memories. The local versions of our codes in 3D have no string-like
logical operators. To take advantage of this for self-correction, we need a local Hamiltonian
that has the code space as the (at least quasi-) degenerate ground space and a favorable
spectrum.
The underlying code should also have a threshold against random errors [1]}.
The obvious choice of Hamiltonian is minus the sum of the gauge generators, but
this will not be gapped in general. Indeed, the simplest example of a Clifford circuit – a wire of
identity gates – maps directly onto the quantum \(XY\)  model, which is gapless when the
coupling strengths are equal [2]}, but somewhat encouragingly is otherwise
gapped and maps onto Kitaev's proposal for a quantum wire [3]}. Other models
of subsystem code Hamiltonians exist; some are
gapped [4]}, [5]}, [6]} and some are
not [7]}, [8]}. Addressing the lack of a satisfying general theory of gauge
Hamiltonians is perhaps a natural first step in trying to understand the power of our
construction in the quest for a self-correcting memory.
